In spite of Aki's historical revelations about the relationship between
Godel and Robinson, the context of the remark Jon Barwise quoted from Vol
III of the Godel papers makes it fairly clear that 'infinitesimal
calculus' just means (as it used to do) the (differential and integral)
calculus. The next paragraph shows that Godel was refering to the 19th
century. The foundations he was refering to was the epsilon-delta
foundations.
There really does not seem to have been that much support for the
existence of infinitesimals---prior, that is, to Robinson's work. Leibniz
seems not to have believed in them---or better, in mathematics he
regarded them as a dispensable computational convenience, to be
understood as `syncategorematic infinites' (i.e. epsilon-delta more or
less). The 18th century mathematicians seem also to have taken this view.
Both Bolzano and Cantor, in locating the concept of (categorematic)
infinity as applying to sets, repudiated the notion of infinitesimal.
Cantor is quoted by Dauben as saying that Johannes Thomae (who had an
office down the hall from Frege) was the first to ``infect mathematics
with the Cholera-Bacillus of infinitesimals''.
May we all show such decent restraint in dealing with disagreements.
Bill Tait
P.S. Moshe': What does OTT mean?